Simplify the following expression: $y = \dfrac{9x^2+26x+16}{9x + 8}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(9)}{(16)} &=& 144 \\ {a} + {b} &=& &=& {26} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $144$ and add them together. The factors that add up to ${26}$ will be your ${a}$ and ${b}$ When ${a}$ is ${8}$ and ${b}$ is ${18}$ $ \begin{eqnarray} {ab} &=& ({8})({18}) &=& 144 \\ {a} + {b} &=& {8} + {18} &=& 26 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({9}x^2 +{8}x) + ({18}x +{16}) $ Factor out the common factors: $ x(9x + 8) + 2(9x + 8)$ Now factor out $(9x + 8)$ $ (9x + 8)(x + 2)$ The original expression can therefore be written: $ \dfrac{(9x + 8)(x + 2)}{9x + 8}$ We are dividing by $9x + 8$ , so $9x + 8 \neq 0$ Therefore, $x \neq -\frac{8}{9}$ This leaves us with $x + 2; x \neq -\frac{8}{9}$.